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**On the definition of the Galois groupoid.**
*(English.
French summary)*
Zbl 1205.12005

Cano, Felipe (ed.) et al., Équations différentielles et singularités. En l’honneur de J. M. Aroca. Paris: Société Mathématique de France (ISBN 978-2-85629-263-1/pbk). Astérisque 323, 441-452 (2009).

A Galois theory for linear differential equation has existed ever since Picard and Vessiot; it has been put in rigorous and modern form by Ritt and Kolchin and has seen many developments in the last thirty years. As for nonlinear equations, the story has been slower, partly because of the difficulty of extending the theory of algebraic groups to infinite dimension, partly because of the many technical difficulties proper to nonlinear equations.

In [Nagoya Math. J. 144, 59–135 (1996; Zbl 0878.12002)] the author of the paper under review succeeded in defining such a Galois group, but his theory has not been very much used and developed. In [Monogr. Enseign. Math. 38, 465–501 (2001; Zbl 1033.32020)] B. Malgrange introduced “D-groupoids” and showed how to use them to build a Galois theory. The theory of Malgrange has benefitted all the (rather heavy) apparatus of differential geometry, partly along the ideas of Élie Cartan; it has been successfully used to solve longstanding problems.

In the present paper, Umemura explains why his theory and Malgrange’s are, in essence, equivalent. This is not a formal proof, but a nice pedagogical explanation of both theories and a description of the bridge between them on a significant example.

Complete proofs will appear elsewhere and have been announced in [C. R., Math., Acad. Sci. Paris 346, No. 21-22, 1155–1158 (2008; Zbl 1204.12009)] under the title ”Sur l’équivalence des théories de Galois différentielles générales”.

For the entire collection see [Zbl 1187.00035].

In [Nagoya Math. J. 144, 59–135 (1996; Zbl 0878.12002)] the author of the paper under review succeeded in defining such a Galois group, but his theory has not been very much used and developed. In [Monogr. Enseign. Math. 38, 465–501 (2001; Zbl 1033.32020)] B. Malgrange introduced “D-groupoids” and showed how to use them to build a Galois theory. The theory of Malgrange has benefitted all the (rather heavy) apparatus of differential geometry, partly along the ideas of Élie Cartan; it has been successfully used to solve longstanding problems.

In the present paper, Umemura explains why his theory and Malgrange’s are, in essence, equivalent. This is not a formal proof, but a nice pedagogical explanation of both theories and a description of the bridge between them on a significant example.

Complete proofs will appear elsewhere and have been announced in [C. R., Math., Acad. Sci. Paris 346, No. 21-22, 1155–1158 (2008; Zbl 1204.12009)] under the title ”Sur l’équivalence des théories de Galois différentielles générales”.

For the entire collection see [Zbl 1187.00035].

Reviewer: Jacques Sauloy (Toulouse)